Physics 211B : Problem Set #1
[1] Recall that a quantity F (k) that is conserved between collisions satisfies
d F (r) =
dt
Z
d3k
F (k)
(2π)3
∂f
∂t
.
k,coll
Ω̂
In this context, we see how the relaxation time approximation,
∂f
∂t
=−
k,coll
δf (r, k, t)
,
τ (k)
where δf = f − f 0 , violates number conservation. To remedy this situation, one can
orthogonalize the relaxation approximation collision integral to this collisional invariant.
This is the so-called BTK collision integral, first discussed in P. L. Bhatnagar, E. P. Gross,
and M. Krook, Phys. Rev. 94, 511 (1954):
Tr f /τ
f
f0
∂f
·
=− +
,
∂t k,coll
τ
τ
Tr f 0 /τ
where Tr ↔
R
d3k
(2π)3
. How would you write down a linearized collision integral which would
Ω̂
preserve both particle number and energy?
[2] In eqns. 1.91 – 1.97 of the notes, I write the linearized Boltzmann equation with E = 0
in terms of the amplitudes ALM in the different angular momentum channels. Work out
the equation corresponding to 1.95 when E is included. Show that the coefficients of the
spherical harmonics all decay to zero except for the case L = 1. How do you identify the
transport lifetime?
[3] Thoroughly study problem #5 from the example problems for chapter 1. Compute the
real part of σxx (B, ω) as a function of field amplitude B for the direction B̂ and frequency
given in the problem, both for Si and Ge. Assume different values for ωτ and find the value
of τ which best fits the data in each case.
[4] The spin-orbit Hamiltonian is
HSO =
~
σ · ∇V × p .
4m2e c2
Write down the Boltzmann equation for scattering within a single band, treated within the
effective mass approximation, for a collection of randomly distributed but otherwise identical
spin-orbit scatterers. You should only consider spin-orbit scattering in this problem (a
highly artificial situation) and neglect any potential scattering. You should derive coupled
1
Boltzmann equations for δfk↑ and δfk,↓ . It will make your life easier if, for each k, you
choose k as the quantization axis for the spin. Writing
δfk,s = 12 δfk,↑ + δfk,↓
δfk,a = 12 δfk,↑ − δfk,↓ ,
and then taking
δfk,α =
X
ALM α (k, t) YLM (k̂) ,
L,M
where α = s or a, derive equations for the rate of change of the coefficients ALM α in the
absence of any external fields.
2